Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers

This paper realizes a symbiosis of two approaches to the numerical solution of second order ODEs with a small parameter having singularities such as interior and boundary layers, namely, the application of both compact schemes of high orders and layer-resolving grids. The generation of layer-resolving grids, based on estimates of solution derivatives and formulations of coordinate transformations eliminating solution singularities, is a generalization of the methodology early developed for the first order scheme.
This paper presents the formulas of the coordinate transformations and numerical experiments for the schemes of the first, second, and third orders on uniform and layer-resolving grids for the equations with boundary, interior, exponential and power layers of the first and second scales. The experiments conducted confirm the uniform convergence of the numerical solutions of equations with the help of compact schemes of high orders on the layer-resolving grids.
By using the transfinite interpolation methodology or numerical solutions to the Beltrami and diffusion equations in a control metric, built by the coordinate transformations eliminating the solution singularities, the developed technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers.